Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(10,13)(11,15)(17,18,20,21)(19,23,24,22)(25,26)(27,28)(29,30)(31,32);(1,3,5,8)(2,4,6,7)(11,15)(14,16)(18,21)(19,24)(26,30)(28,32);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24)(25,27,29,31)(26,28,30,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24)(25,28,29,32)(26,27,30,31);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232584, #72232610, #72232630, … |
| maximal normal subgroup of | #72232584, #72232610, #72232630, … |
| maximal quotient of | #72329930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3964421);